This Demonstration presents a random oriented graph that has at least one terminal vertex. Such graphs can represent the terminal game Nim. To play Nim, players alternate in choosing a sequence of connected vertices until a terminal vertex is reached by one of the players; then the other player wins.

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Details

• a function (i.e. for every element , there is a vector ); is called the payoff function and each player tries to maximize it.

An arbitrary oriented graph with at least one terminal vertex can represent a Nim-like game that is a particular case of a terminal game on graphs with , , and equal to the set of terminal vertices. It can be verified that can be partitioned into subsets and (i.e. and ) with the properties:

• ,

• .

The vertices in the set are called the "gain" vertices, and those in are called the "loss" vertices, because the player who chooses a vertex from offers only losing vertices to the opponent (from ), and the player who chooses a vertex from offers only winning vertices (from ). Obviously is the graph's kernel. Define to be the set of terminal vertices.